... filters,1
Following previous usage conventions, we use the term ladder filter'' when the graph of the filter is planar, and lattice filter'' when the graph is non-planar.
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... tube,2
Stiffness'' is defined here for air as the reciprocal of the adiabatic compressibility of the gas [61, p. 230]. This definition helps to unify the scattering formalism for acoustic tubes with that of mechanical systems such as vibrating strings.
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....3
There are no tube shapes supporting exact traveling waves other than cylindrical and conical (or conical wedge, which is a hybrid) [69]. However, the Salmon horn family'' (see, e.g., [60,96]) characterizes a larger class of approximate one-parameter traveling waves.'' In the cone, the wave equation is solved for pressure using a change of variables , where is the distance from the apex of the cone, causing the wave equation for the cone to reduce to that of the cylindrical case [2].
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... power4
Note that corresponds to the average physical power at frequency , where , and the wave variable magnitudes on the unit circle may be interpreted as RMS levels [4, p. 48]. For , we may interpret the power as the steady state power obtained when exponential damping is introduced into the waveguide giving decay time-constant , where [4, p. 48].
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....5
A complex-valued function of a complex variable is said to be positive real if
1)
2)
Positive real functions characterize passive impedances in classical network theory [129].
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...6
Note that by the maximum modulus theorem, if a matrix of meromorphic functions is positive definite on the unit circle and analytic outside the unit circle, it is necessarily positive definite everywhere outside the unit circle.
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... velocity7
The symbols for the variables velocity and force have been chosen to maintain consistency with the analogous acoustical quantities.
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... sample,8
Strictly speaking, a squared wave variable in a normalized waveguide is proportional to signal power [97], and we should multiply by the sampling interval to obtain units of energy. However, such a constant scale factor is inconsequential.
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...WAVCAS.9
Altering a ladder filter to convert it to a DWN can also be accomplished using the cut-set method of Kung [51,12].
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